A322913 Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).

1, 3, 7, 15, 36, 81, 197, 455, 1105, 2618, 6315, 15141, 36570, 88161, 213342, 516247, 1251728, 3037059, 7378290, 17938430, 43655465, 106317863, 259127707, 631986437, 1542364386, 3766351332, 9202390342, 22496047757, 55020807236, 134631987776, 329579227722, 807142635031

Offset

1,2

Comments

The sequence (A032173(n): n >= 1) shifts two places to the left under Bower's "CHK" (necklace, identity, unlabeled) transform. The current sequence satisfies A032173(n+2) = (1/n)*Sum_{d|n} mu(d)*a(n/d).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

C. G. Bower, Transforms (2)

Formula

a(n) = Sum_{d|n} d*A032173(d+2).

a(n) = n*A032173(n) + Sum_{s=1..n-1} a(s)*A032173(n-s).

G.f.: If A(x) = Sum_{n>=1} a(n)*x^n and B(x) = Sum_{n>=1} A032173(n)*x^n, then A(x) = x*(dB(x)/dx)/(1-B(x)), while (B(x) - x - x^2)/x^2 = Sum_{n>=1} A032173(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-B(x^n)).

Mathematica

(* b = A032173 *) b[1] = b[2] = 1; c[1] = 1; c[2] = 3;
b[n_] := b[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
c[n_] := c[n] = n b[n] + Sum[c[s] b[n-s], {s, 1, n-1}];
a[n_] := Sum[d b[d+2], {d, Divisors[n]}];
Array[a, 26] (* Jean-François Alcover, Jan 02 2019 *)

Prog

(PARI)
CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x)); for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(deriv(x*p)/(1-x*p)+O(x^n))} \\ Andrew Howroyd, Apr 27 2020

Crossrefs

Cf. A032171, A032173, A032174.

Sequence in context: A081669 A086821 A007576 * A167539 A223167 A335780

Adjacent sequences: A322910 A322911 A322912 * A322914 A322915 A322916

Keyword

nonn

Author

Petros Hadjicostas, Dec 30 2018

Extensions

Terms a(27) and beyond from Andrew Howroyd, Apr 27 2020

Status

approved